The dimension of a poset is the least integer such that the poset is isomorphic to a subposet of the product of linear orders. In 1983, Kelly constructed planar posets of arbitrarily large dimension. Crucially, the posets in his construction involve large standard examples, the canonical structure preventing a poset from having small dimension. Kelly’s construction inspired one of the most challenging questions in dimension theory: are large standard examples unavoidable in planar posets of large dimension? We answer the question affirmatively by proving that every -dimensional planar poset contains a standard example of order . More generally, we prove that every poset from Kelly’s construction appears in every poset with a planar cover graph of sufficiently large dimension.
joint work with Heather Smith Blake, Jędrzej Hodor, Piotr Micek, and William T. Trotter.
